# Rational Krylov for Stieltjes matrix functions: convergence and pole selection

## TL;DR

This paper develops effective pole selection strategies for rational Krylov methods to efficiently compute matrix functions, especially for Cauchy- and Laplace-Stieltjes functions, with applications to tensor-structured problems.

## Contribution

It introduces new pole selection techniques with convergence bounds for rational Krylov methods applied to Stieltjes matrix functions, including tensor-structured cases.

## Key findings

- Effective pole selection strategies with explicit convergence bounds.
- Application to tensor-structured matrices in fractional diffusion problems.
- Leverages Kronecker structure for efficient computation.

## Abstract

Evaluating the action of a matrix function on a vector, that is $x=f(\mathcal M)v$, is an ubiquitous task in applications. When $\mathcal M$ is large, one usually relies on Krylov projection methods. In this paper, we provide effective choices for the poles of the rational Krylov method for approximating $x$ when $f(z)$ is either Cauchy-Stieltjes or Laplace-Stieltjes (or, which is equivalent, completely monotonic) and $\mathcal M$ is a positive definite matrix. Relying on the same tools used to analyze the generic situation, we then focus on the case $\mathcal M=I \otimes A - B^T \otimes I$, and $v$ obtained vectorizing a low-rank matrix; this finds application, for instance, in solving fractional diffusion equation on two-dimensional tensor grids. We see how to leverage tensorized Krylov subspaces to exploit the Kronecker structure and we introduce an error analysis for the numerical approximation of $x$. Pole selection strategies with explicit convergence bounds are given also in this case.